Insert using LEFT JOIN and INNER JOIN

You have to be specific about the columns you are selecting. If your user table had four columns id, name, username, opted_in you must select exactly those four columns from the query. The syntax looks like:

INSERT INTO user (id, name, username, opted_in)
SELECT id, name, username, opted_in 
FROM user 
LEFT JOIN user_permission AS userPerm ON user.id = userPerm.user_id

However, there does not appear to be any reason to join against user_permission here, since none of the columns from that table would be inserted into user. In fact, this INSERT seems bound to fail with primary key uniqueness violations.

MySQL does not support inserts into multiple tables at the same time. You either need to perform two INSERT statements in your code, using the last insert id from the first query, or create an AFTER INSERT trigger on the primary table.

INSERT INTO user (name, username, email, opted_in) VALUES ('a','b','c',0);
/* Gets the id of the new row and inserts into the other table */
INSERT INTO user_permission (user_id, permission_id) VALUES (LAST_INSERT_ID(), 4)

Or using a trigger:

CREATE TRIGGER creat_perms AFTER INSERT ON `user`
FOR EACH ROW
BEGIN
  INSERT INTO user_permission (user_id, permission_id) VALUES (NEW.id, 4)
END

INSERT INTO Test([col1],[col2]) (
    SELECT 
        a.Name AS [col1],
        b.sub AS [col2] 
    FROM IdTable b 
    INNER JOIN Nametable a ON b.no = a.no
)

you can't use VALUES clause when inserting data using another SELECT query. see INSERT SYNTAX

INSERT INTO user
(
 id, name, username, email, opted_in
)
(
    SELECT id, name, username, email, opted_in
    FROM user
         LEFT JOIN user_permission AS userPerm
            ON user.id = userPerm.user_id
);